Lecture No 1 Introduction to Diffusion equations The heat equat

Transcription

1 Lecture No 1 Introduction to Diffusion equations The heat equation Columbia University IAS summer program June, 2009

2 Outline of the lectures We will discuss some basic models of diffusion equations and present their basic properties. Such models include various geometric flows such as: the curve shortening flow, the Ricci flow on surfaces and the Yamabe flow. We will discuss existence, regularity, a priori estimates, asymptotic behavior and classification of solutions. Lecture 1: Introduction and the heat equation Lecture 2: Slow diffusion and free-boundaries Lecture 3: Fast diffusion and the Ricci flow on surfaces Lecture 4: Ancient solutions to the curve shortening flow and the Ricci flow on surfaces

3 Introduction to diffusion The simplest model of linear diffusion is the familiar heat equation: u t = u, u = u(x, t), x R n, T > 0 where u := n i=1 u x i x i = div ( u). (Diffusion of heat, chemical concentration). One of the simplest models of non-linear diffusion is the porous medium (m > 1) or fast-diffusion (m < 1) equation u t = u m, u 0, x R n, T > 0. (Diffusion of gas through a porous medium, population dynamics, gas kinetics, diffusion in plasma, geometry). Another simple model is the reaction-diffusion equation u t = u + u p, u 0, p > 0. (Population dynamics, geometry).

4 Other models of Diffusion Evolution p-laplacian Equation (quasi-linear) u t = div ( u p 2 u), p > 0. Motion of a curve y = u(x, t) by its curvature u xx u t = 1 + ux 2. Motion of a surface z = u(x, y, t) in R 3 by its mean curvature u t = (1 + u2 y )u xx 2u x u y u xy + (1 + u 2 x)u yy 1 + u 2 x + u 2 y Motion of a surface y = u(x, y, t) in R 3 by its Gaussian curvature (fully-nonlinear) u t = det D 2 u (1 + u 2 x + u 2 y ) 3/2.

5 Slow and fast-diffusion Consider the equation u t = u m. Since u m = div ( u m ) = div (m u m 1 u) the diffusion is governed by D(u) := m u m 1. m = 1: The diffusivity D(u) := 1 is constant in any direction and the equation is linear. m > 1: Since the diffusivity D(u) := m u m 1 0, as u 0 and we have slow diffusion. m < 1: Since the diffusivity D(u) := m u m 1 +, as u 0 and we have fast diffusion. m = 0: The equation takes the form u t = log u. m < 0: We have super-fast diffusion. Remark: We will see in these lectures that slow and fast diffusion have very different properties.

6 The Heat Equation - Derivation The heat equation u t = u describes the distribution of heat in a given region over time. The heat equation can be derived from the following principles: The amount of heat Q contained in a region Ω is proportional to the temperature T, the density ρ and the heat capacity κ s of the material, i.e. if Ω is a region of the material: Q = ρ κ s T (x, t) dx. (1) Ω The heat transfer through the boundary Ω of the region is proportional to the heat conductivity σ, to the gradient of the temperature across the region and to the area of contact, i.e. dq dt = Ω σ T n da. (2)

7 The Heat equation - Derivation If we differentiate (1) in time and apply the divergence Theorem in (2) we obtain: T ρ κ s Ω t dx = div (σ T ) da. Ω Since Ω can be an arbitrary part of the material under study, we obtain the heat equation T t = λ T if we assume that ρ, κ s and σ are independent of the position x. Remark: If the heat conductivity σ is taken to depend on the temperature T, then we obtain non-linear versions of the heat equation.

8 Parabolic scaling and the Fundamental Solution Parabolic Scaling: If u(x, t) solves the heat equation, then ũ(x, t) = u(γ x, γ 2 t), γ > 0 also solves the heat equation. Self-Similar solution: The above scaling suggests that we search for a special radially symmetric solution of the form ( ) x 2 Φ(x, t) = α(t) v. t for some functions α(t) and v(r). The above representation leads to the fundamental solution 1 Φ(x, t) = (4πt) n 2 e x 2 4t, t > 0.

9 Properties of the Fundamental solution A simple calculation leads to: R n Φ(x, t) dx = 1, t > 0. Also, lim t 0 Φ(x, t) = 0, for all x 0. Hence, lim Φ(, t) = δ 0. t 0 We observe the following two characteristic properties of the fundamental solution: Infinite speed of propagation: Φ(, t) > 0, for all t > 0. Smoothing effect: Φ is C smooth in x and t, for all t > 0. Remark: All solutions of the heat equation have the above two properties. Non-linear equations don t in general.

10 The Cauchy problem Solutions u of the Cauchy problem { u t = u in R n [0, T ) ( ) u(, 0) = g on R n are given by 1 u(x, t) = Φ(x y, t) g(y) dy = R n (4πt) n 2 R n e x y 2 4t Indeed, we have: u t x u = [(Φ t x Φ)(x y, t)] g(y) dy = 0 R n and (since lim t 0 Φ(x y, t) = δ x (y)): lim u(x, t) = lim Φ(x y, t) g(y) dy = g(x). t 0 t 0 R n g(y) dy.

11 Non-homogeneous problem Consider now solutions u of the non-homogeneous problem { u t u = f in R n (0, ) ( ) u(, 0) = 0 on R n. Let u(x, t; s) be the solution of the Cauchy problem { u t = u in R n (s, ) u(, s) = f (, s) on R n Duhamel s principle asserts that the solution u of ( ) is given by: Hence, u(x, t) = u(x, t) = t t 0 0 u(x, t; s) ds, x R n, t 0. R n Φ(x y, t s) f (y, s) dy ds.

12 The Mean-value formula The well known mean value formula for harmonic functions: u = 0 in Ω R n, asserts that 1 u(x 0 ) = u(x) dx, if B r (x 0 ) Ω. B r (x 0 ) B r (x 0 ) A similar formula holds for solutions of the heat equation u t = u in Q T = Ω (0, T ]. Considering the parabolic balls: { E(x 0, t 0 ; r) := (x, t) R n+1 t t 0, Φ(x x 0, t 0 t) 1 } r n then u(x 0, t 0 ) = 1 4r n if E(x 0, t 0 ; r) Q T. E(x 0,t 0 ;r) u(x, t) x x 0 2 dx dt (t t 0 ) 2

13 The Strong Maximum Principle The parabolic boundary p Q T of the cylinder Q T := Ω (0, T ] is defined as: p Q T = ( Ω (0, T ]) (Ω {0}). Strong maximum principle: If u solves the heat equation in Q T then: We have: max u = max u. Q T pq T Furthermore, if Ω is connected and there exists a point (x 0, t 0 ) Q T such that u(x 0, t 0 ) = max u Q T then u must be constant in Q T.

14 Uniqueness of Solutions Uniqueness on bounded domains: If g C( p Q T ), then there exists at most one solution u C 2,1 (Q T ) C( Q T ) of: { u t = u in Q T u = g on p Q T Maximum principle for the Cauchy problem: If u solves { u t = u in R n (0, T ) u(, 0) = g on R n u C 2,1 (R n (0, T ]) C(R n [0, T ]) and satisfies the growth estimate u(x, t) A e a x 2, then sup R n [0,T ] u = sup R n g. Uniqueness: Solutions to the Cauchy problem which satisfy the above growth condition are unique. Remark: The growth assumption is necessary!

15 Regularity Let Q r (x 0, t 0 ) denote the parabolic cylinder of size r around a point (x 0, t 0 ), namely: Q r (x 0, t 0 ) := B r (x 0 ) [t 0 r 2, t 0 ]. Derivative estimates: For each k, l = 0, 1,, there exists a constant C k,l such that max Q r 2 (x 0,t 0 ) Dk x D l tu C k,l r k+2l+n+2 u L 1 (Q r (x 0,t 0 )) for all cylinders Q r (x 0, t 0 ) Q T and all solutions to the heat equation in Q T. Conclusion: Local solutions to the heat equation are C -smooth. Proof: You use the exact representation of solutions to the heat equation in terms of the initial data and cut-off functions.

16 Schauder Estimate Parabolic Hölder spaces: We say that u C α (Q T ) if u(x 1, t 1 ) u(x 2, t 2 ) C ( x 1 x 2 + t 1 t 2 ) α. We say that u C 2+α (Q T ) if u, u t, D x u, Dx 2 u C α (Q T ). Schauder estimate: If u is a smooth solution of { u t u = f in Q T u = g on p Q T then u C 2+α (Q T ) C ( u C 0 (Q T ) + f C α (Q T ) + g C 2,α ( Q T )).

17 Harnack Inequality Assume that u is a nonnegative solution of the heat equation u t = u, t > 0. Then, u satisfies the Harnack inequality: u(x 1, t 1 ) ( t2 t 1 for all 0 < t 1 < t 2 and x 1, x 2 R n. ) n 2 u(x2, t 2 ) e x 2 x 1 2 4(t 2 t 1 ) Application: If u(0, T ) M <, then for all t < T ɛ we have u(x, t) C(M, T ) t n x 2 2 e 4ɛ i.e. nonnegative solutions of the heat equation grow at most exponentially as x.

18 Widder theory for the heat equation Let u be a nonnegative distributional solution of the heat equation u t = u in S T := R n (0, T ]. The initial trace µ exists; there exists a nonnegative Borel measure µ 0 on R n such that lim t 0 u(, t) = dµ 0 and satisfies the growth condition ( ) e C x 2 T dµ 0 < where C is an absolute constant. The trace µ 0 determines the solution uniquely. For each nonnegative Borel measure µ 0 on R n satisfying the ( ), there is a nonnegative continuous weak solution u the heat equation in S T with trace µ 0 and u(x, t) = C n t n/2 e x y 2 4t dµ 0 (y) for an absolute constant C n depending only on dimension n.